IV. When three lines are in continued proportion, if one be given and the difference of the other two, those other two may be found.

1o. Given the mean BD, and AC the difference of the extremes.

3 C. & D. 2 In As ADB, DCB we have BDC= /DAC, and B com. 4 Def. 1 VI.. As ADB, DCB, are eq. ang., and their sides proportional; 54, VI. AB: DB DB : BC.

2o. Given one extreme, BC, and the difference between the mean and the other extreme, AB-BD; to find both the mean and that other extreme.

D. 1Theo. I.Usel. the differences of successive terms are as the terms them

selves,

.. the rect. under one extr. and the dif. of the mean and the other extr. the rect. under the mean and the difference between it and the given extreme.

i. e., BC. (AB-DB): DB. (DB--BC).

Since the Area of this rect. and the difference of the sides are given;

..the sides themselves may be found.

For√(difference) 2

5 Use 3,

10, II.

6 Remk.

Sum.

+the area =

2

2

LARDNER'S Euclid, p. 79.

Sum Diff.

V. To find two st. lines to contain a rectangle equal to a given rectangle AB. BC, and to have a given ratio one to the other.

D. 1 C. 2, 3. 22, VI.

3

a mean propl.,

i. e., BC BE =
BE: BD,

and through E draw EF

|| DA, meeting AB

in F;

A

F

B

the lines BE, BF, are the st. lines required.

Now the lines are evidently in the given ratio;
and. EF || DA, ... AB : BD = FB : BE;
and.. also BF: BABE : BD;

4 20, VI... BF. BE: BA.BD

C. 3.

51, VI. 69, V. 7 Sol.

i. e. BC: BD;

BE2: BD2,

but AB. BC: AB. BD = BC: BD,

EB. BF: = AB. BC.

BE and BF are the two required fines.

Q. E. F.

D

VI. We are also able by the 13th Prop. to find any number of means represented by a power of 2, minus 1; i. e. 4-1 or 3, 8-1 or 7, 16-1 or 15 &c. For having obtained one mean between two magnitudes, we may by the same process determine a mean between it and each of its extremes, and thus we shall have three means. And again between each successive pair of the series thus found inserting means, and pursuing the same method, we obtain seven means, fifteen, thirty one, &c.

ADDENDA to PROP. 13, VI.

I.-The Problem, to obtain two mean proportionals between two given st. lines, on which depends "the duplication of the cube" in Solid Geometry, cannot be determined by the rule and compasses, the only instruments allowed by EUCLID; and consequently it has never received a strictly Geometrical Solution. Various mechanical contrivances have been invented to attain the solution; as PLATO'S Method of a ruler inserted perpendicularly on the side of a square and moveable along that side; PHILO's of Byzantium, of a graduated ruler revolving on a point; the Trammel of APOLLONIUS, or of NICOMEDES who lived

about two centuries before the birth of Christ; and the Method of DES CARTES, consisting of a collection of rulers, two of which move round a pivot, each of the two having a groove on the inner edge, in which other rulers are free to move perpendicularly to the groove. For a description of these Methods reference may be made to LARDNER'S Euclid pp 196-199 COOLEY'S Supplement to Euclid pp 92, 93; or Galbraith's Manual pp. 111, 112. There are several solutions in the third Bk. of the Mathematical Collections of PAPPUS, who flourished A.D. 379-395; and ten different solutions occur in a Commentary by EUTOCIUS of Ascalon, who lived AD., 560, on the Sphere and Cylinder of ARCHIMEDES.

1o. PLATO'S Method of a perpendicular moveable along the side of a Square.

C. 1

2

3

5

י7

D. 1 C.

2 8, VI. 3 Conc.

Take two st. rulers AB, BC, fitted
at a rt. ABC;

and a third ruler moving along
AB at rt. s to AB;

place the given extremes FG and
GH at a rt FGH;

and so that FG produced cuts the
rt. ABC in the vertex B ;

while HG produced cuts BA
in D,

E E

C

H

IN

and the perp. DE meets the . s D & F;

B

then GB and GD are the two mean proportionals to FG & GH.

... HBD & FDB are rt. d As of which the altitudes
are GB, GD;

HG: GB: GD, and BG : GD : GF.

GB & GD are mean proportionals to the extremes
HG & GF.

2o. PHILO'S Method of a graduated ruler FG revolving round, B, the vertex of a rt. angle.

2 36, III.

3' Ax. 1.

to AB & BC.

.. BG

=

QF, & QG BF; .. rect. QG. GB =

rect. QF. BF;

but QG. GB = DG. GC ; & QF. BF = DF. FA;
.. DG. GC DF. FA;

=

.. DG: DF = AF: GC;

but. As GDF, BAF & GCB are equiangular,
.. DG : DF = AB : AF, & DG : DF = GC : CB;
and .. BA: AF

AF: GC;

3°. Method of DES CARTES, with a collection of rulers.

d f h k

A

HK M C

Take two st. rulers AB, BC united by a pivot at B ;
in each there is a groove, in which rulers move
perpendicularly to AB & BC;

so that perp. Dd, on opening the rulers, pushes forward E e,
Ee pushes Ff &c.

Let two means be required.

Move Dd from B, until BD = the less extreme;

=

close AB upon CB that the perpendicular may move up to D,
and fix D d in the position where BD the less extreme;
Now open AB & CB, until BG the greater extreme,
Dd, during the process, pushing Ee from B, & E e pushing
Ff, &c.;

then the two means are BF & BE.

As BDE & BFG are equiangular;

BD BF BF : FE; & BF : FE = FE : BG ;
BF & BE are the two means to BD & BG

In a similar way three means BE, BF, BG will be found to BD & BI, by opening the rulers AB, CB, until BH

=

the greater extreme.

For four mean proportionals open the rulers until BI

and so on for a greater number of means.

Generally, if an even number of means be required, the extremes will be on different rulers; if an odd number, on the same ruler.

N.B.-The advantage of the Method of DES CARTES is more apparent than real. "It would appear that any number of mean proportionals may be found; but the necessary adjustments are almost as deficient in practical facility as in geometrical legitimacy."-COOLEY.

II. The Trisection of a rectilineal angle, ABC, is another puzzle for Plane Geometry. The Solution of this Problem, though of no more trouble by aid of the Higher Analysis than the finding of the cube root, is impossible by the circle and right line. NICOMEDES, known as the discoverer of the conchoid curve, effected the Solution by the Trammel which he invented, and by which the Duplication of the Cube and the Trisection of an angle are both accomplished.

1o. The TRAMMEL of NICOMEDES, a T square with a moveable ruler HG.

C. 1

21

345

6789

10

Take a T square with a groove CD, in the cross ruler AB;
and a fixed pin, I, in the stem, FE;

Let this fixed pin be inserted in the groove of the moveable
ruler HG;

and HG also have a fixed pin K, inserted in CD.

From H, in HG, there issues a sliding stem HP;

Its length HP = that part of the line to be intercepted by the
sides of the angle.

Let the fixed pin I be now placed on the given point;

and the groove CD on one side of the given angle ;

let HG be moved so that the pin K go along one side of the angle;
and let the motion be continued until the P shall come

on the other side of the angle;

then the line joining P & I, i. e., PI, is the required line.

20. By this instrument to Trisect a given angle ABC.